A390468 Average of the number of "ON" cells at n-th stage in the "Ulam-Warburton" two-dimensional cellular automaton (A147562) and the number of toothpicks at n-th stage in the "Toothpick" two-dimensional automaton (A139250).

0, 1, 4, 8, 16, 20, 30, 42, 64, 68, 78, 90, 114, 128, 160, 194, 256, 260, 270, 282, 306, 320, 352, 386, 450, 464, 496, 532, 606, 654, 752, 846, 1024, 1028, 1038, 1050, 1074, 1088, 1120, 1154, 1218, 1232, 1264, 1300, 1374, 1422, 1520, 1614, 1794, 1808, 1840, 1876, 1950, 1998, 2096, 2192, 2382, 2430, 2530, 2640, 2872

Offset

0,3

Comments

It appears that the graph is very close to the graph from the Flower garden sequence (A323650), (see the conjecture 2 and also the Plot 2 in the Links section).

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191, which is also available at arXiv:1004.3036v2

Steven R. Finch, Toothpicks and Live Cells, July 21, 2015. [Cached copy, with permission of the author]

OEIS Plot 2, A390468 vs A323650

Formula

a(n) = (A147562(n) + A139250(n))/2.

Conjecture 1: a(2^k) = 4^k, k >= 0.

Conjecture 2: a(2^k) = 1 + A323650(2^k), k >= 1.

Mathematica

A139250[n_] := A139250[n] = If[n <= 1, n, (2*#^2 + 1)/3 + 2*A139250[n - #] + A139250[n - # + 1] - 1 & [2^(BitLength[n] - 1)]]; (* After David A. Corneth at A139250 *)
A147562list[nmax_] := Accumulate[Join[{0, 1}, 4*3^(DigitCount[Range[nmax - 1], 2, 1] - 1)][[;; nmax + 1]]];
A390468list[nmax_] := (A147562list[nmax] + Array[A139250, nmax + 1, 0])/2;
A390468list[100] (* Paolo Xausa, Dec 16 2025 *)

Crossrefs

Cf. A261313 is twice A195853.

Cf. A139250, A147562, A323650.

Sequence in context: A092259 A312805 A312806 * A036693 A381837 A181471

Adjacent sequences: A390465 A390466 A390467 * A390469 A390470 A390471

Keyword

nonn

Author

Omar E. Pol, Nov 06 2025

Status

approved